Cycle Ambiguity Reacquisition in UAV

Applications Using a Novel GPS/INS Integration

Algorithm

Steven E. Langel, Samer M. Khanafseh, Fang-Cheng Chan, and Boris S. Pervan

Illinois Institute of Technology, Chicago, IL

BIOGRAPHY

Steven Langel is a PhD candidate at the Illinois Institute

of Technology. His work focuses on the design of high

accuracy, high integrity navigation algorithms for

aerospace applications. His work also focuses on the

integration of inertial navigation systems with GPS.

Steven received his B.S degree in aerospace and

mechanical engineering from the Illinois Institute of

Technology.

Samer Khanafseh is a research associate at the Illinois

Institute of Technology whose work involves high

accuracy and high integrity navigation system design. In

addition, he performs fundamental research in the field of

cycle ambiguity resolution with constrained integrity. Dr.

Khanafseh received his PhD degree in mechanical and

aerospace engineering from the Illinois Institute of

Technology.

Fang-Cheng Chan is a research associate at the Illinois

Institute of Technology. His work focuses on designing

navigation algorithms for high integrity situations such as

precision approach and landing. He also performs

research in the field of GPS/INS integration with

emphasis on developing new approaches to tightly

coupled integration. Dr. Chan received his PhD degree

from the Illinois Institute of Technology.

Boris Pervan is an associate professor of aerospace

engineering at the Illinois Institute of Technology. His

work pervades many fields of navigation research

including: carrier phase DGPS, aircraft precision

approach and landing, optimal and robust navigation

system design, integrity monitoring and integration of

inertial and laser sensors with GPS. Dr. Pervan received

his PhD degree in aeronautics and astronautics from

Stanford University.

ABSTRACT

A novel GPS/INS integration algorithm that is adaptable

to high satellite blockage environments is developed.

Aviation applications such as autonomous shipboard

landing require high degrees of accuracy and integrity

which warrants the use of a carrier phase differential GPS

navigation architecture. In order to achieve the high

positioning accuracy achievable with the carrier phase, it

is necessary to resolve the integer cycle ambiguity. To

include satellites that are acquired (or lost and reacquired)

after the initial fix, a „dual-track‟ fixing method is

introduced. In this algorithm an independent fixing track

is implemented in which constrained-integrity fixes are

continuously attempted with the newly acquired satellites.

The performance of the dual track fixing algorithm is

further enhanced by the integration of an inertial

navigation system (INS) with the GPS carrier phase. In

qualitative terms, during brief satellite outages the

accurate position propagation provided from the output of

the INS will translate into knowledge of the cycle

ambiguities for the reacquired satellite. In this work, the

integration concept is implemented in a novel GPS/INS

mechanization that combines GPS and INS states

(including cycle ambiguity states) in one centralized

Kalman filter. A covariance analysis of the integrated

architecture is carried out for the autonomous shipboard

landing application, and the findings are expressed in

terms of availability. The result of this work is a

navigation architecture capable of providing instrument

landing support for a Case-I approach; a feat as yet

unrealized.

INTRODUCTION

The problem of shipboard landing using the Global

Positioning System (GPS) has been an active area of

research over the past decade. For instance, different

carrier phase differential GPS (CPDGPS) navigation

architectures for shipboard landing of aircraft have been

described in [1], [2] and [3]. A key feature of the dual

frequency algorithm described in [1] is its use of

geometry-free filtering for cycle estimation of widelane

integers. As the approach is initiated, this prior

knowledge is used together with the available satellite

geometric redundancy to fix specific linear combinations

of L1 and L2 cycle ambiguities with high integrity.

Fixing with constrained integrity is performed by means

of a partial fixing integer bootstrap process. This

algorithm was shown to provide good availability

provided that the mission is conducted under clear sky

conditions [1].

One drawback of this algorithm is that there is no

mechanism to regain lost satellites in the event of brief

satellite outages. This issue was addressed in the design

of a navigation system for the aerial refueling mission [4].

In this mission, satellite blockage is introduced from the

refueling tanker body, which degrades the resulting

positioning performance. In order to reacquire lost

satellites without jeopardizing system integrity, a dual

track fixing algorithm is implemented which continuously

attempts constrained-integrity fixes with the newly

acquired satellites. The resulting positioning solution is

therefore always computed using the best available

satellite geometry.

In this work we fuse the concepts of geometry-free pre-

filtering with the dual track fixing algorithm and consider

the problem of shipboard landing in the presence of

satellite blockage. The Case-I recovery is a well

established carrier landing approach that is routinely

executed by Navy pilots and will serve as the mission

profile. During a case-I recovery the aircraft must

complete a series of high banking maneuvers which will

inevitably result in satellite blockage. We will show that

even though we combine the benefits of geometry-free

pre-filtering with the satellite recovery capability of the

dual track fixing algorithm, a GPS navigation architecture

does not result in sufficient system availability.

The performance of the dual track fixing algorithm is

further enhanced by the integration of an inertial

navigation system (INS) with the GPS carrier phase. In

qualitative terms, during brief satellite outages the

accurate position propagation provided from the output of

the INS will translate into knowledge of the cycle

ambiguities for the reacquired satellite. In this work, the

integration concept is implemented in a novel GPS/INS

mechanization that combines GPS and INS states

(including cycle ambiguity states) in one centralized

Kalman filter.

Covariance analysis is the primary simulation tool and is

used to assess navigation system performance.

Subsequent sections will depict performance in terms of

availability on an approach-by-approach basis.

CARRIER LANDING PATTERNS

Aircraft carrier recovery patterns flown by Navy pilots are

quite different depending on the time of day, prevailing

weather conditions and visibility. For example, in

adverse weather conditions and during all night flight

operations the pilot will conduct a case-III approach,

shown in figure 1. Due to the lack of visibility, case-III

recoveries are conducted one aircraft at a time using an

instrument landing system (ILS). On the other hand if the

pilot has good visibility and is returning to the ship during

the daytime, then he/she will conduct a Case-I approach,

shown in figure 2. The advantage of this flight pattern is

that it allows the carrier to recover multiple aircraft during

any given landing mission. Providing instrument landing

support for a Case-I approach is necessary if Unmanned

Aerial Vehicles (UAVs) are incorporated into existing

carrier fleets.

There are six phases of flight that the aircraft must

complete during a case-I recovery. The pilot first enters a

holding pattern (phase 1) until they receive notification

from the ship that they are cleared to land. Upon

receiving clearance, the pilot breaks out of the holding

pattern and descends to „initial‟; the beginning of the final

landing approach (phase 2). A constant altitude fly-by is

then conducted (phase 3) followed by a 180 degree turn at

a high bank angle (phase 4). Finally, the aircraft proceeds

back towards the carrier (phase 5) and completes another

180 degree turn before arriving on the flight deck. It is

important to realize that the pilot will be banking during

this approach which will ultimately result in satellite

blockage.

Since much effort has already been expended in

examining the case-III approach, this work will focus on

designing a navigation algorithm that can support

automatic shipboard landing of UAVs using a case-I

approach.

2,2,22

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x Coordinate of UAV (Nmi)

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Figure 2: Case-I recovery

BASELINE GPS NAVIGATION ARCHITECTURE

Before describing the end state architecture which

includes dual-track fixing and INS, let‟s first consider the

carrier phase differential GPS navigation algorithm

depicted in figure 3. This estimation scheme will be the

basis of comparison for more elaborate navigation

algorithms that will be considered in later sections of this

paper. Prior to the initial fix point, the UAV and ship

filter the geometry free measurements, GFz . These

measurements result in increased observability on

individual L1 and L2 cycle ambiguities, which is

exploited in one snapshot of geometric redundancy. At

this point, specific linear combinations of L1 and L2

integers are fixed with high integrity using the LAMBDA

bootstrap method [5]. The output of this process is a

partially fixed solution

kP that is fed into a Kalman filter

to recursively update the estimation error covariance

matrix.

Figure 3: Baseline GPS navigation architecture with

geometry-free pre-filtering

In this work, we use the following measurement model

for L1 and L2 carrier phase measurements:

(1)

(2)

Where r is the true range to the satellite, sb is the

satellite clock bias, ub is the receiver clock bias, I is the

ionospheric delay on L1, T is the tropospheric delay, 1L

is the L1 carrier wavelength, 1LN is the L1 cycle

ambiguity, 1,Lm is the L1 carrier phase multipath, and

1,L is the L1 carrier phase white measurement noise.

All terms in equation (2) with an L2 subscript are defined

analogously to those given in equation (1).

The satellite clock bias can be eliminated by forming

between-receiver single difference measurements for a

given satellite.

(3)

(4)

1,,1,,1,1

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Where ruur indicates a single difference

operation, e is the line-of-sight unit vector from the

reference receiver to the satellite and urx is the relative

position vector we are trying to estimate.

The differential receiver clock bias urb can be eliminated

by forming between-satellite double difference carrier

phase measurements. Denoting the satellite indices as i

and j, the double difference measurement model is given

by:

(5)

(6)

Where 1Lm and 2Lm are L1 and L2 single difference

multipath states, respectively. In this work, single

difference carrier phase multipath is modeled as a first

order Gauss Markov random process with an associated

time constant, . The terms urI and urT are the residual

ionospheric and tropospheric decorrelation errors,

respectively. Provided that the baseline between the user

and reference receivers is not too large, these errors are

small in comparison to the carrier phase thermal noise.

However, as the user-to-reference separation increases,

decorrelation errors can no longer be neglected and to do

so would jeopardize the robustness of the navigation

architecture. Given the baseline distances involved in the

case-I recovery shown in figure 2, it is necessary to model

the atmospheric decorrelation errors with state

augmentation.

There are many models which can be used to estimate the

tropospheric delay [11]. Since these corrections are not

perfect, some residual error will remain. From the LAAS

accuracy models, we have the following expression for

the tropospheric decorrelation error [6]:

(7)

Where n is the refractivity index of the troposphere, 0h

is the troposphere scale height (taken to be 7000 m) and

h is the height of the user above the reference station.

All satellites can be included in one tropospheric

correction matrix, T. Since the tropospheric decorrelation

error model only depends on the refractivity index, the

matrix T will be a column vector.

(8)

Where m

urT is the tropospheric decorrelation error for the

master satellite.

For the ionosphere, a model is first imposed for the

vertical ionospheric decorrelation error [6]:

(9)

Where vig is the vertical ionospheric gradient for a

given satellite and x is the distance between the UAV

and ship in the x-direction (see figure 2 for a description

of the axes).

The vertical delay given in equation (9) is converted into

actual range delay via an obliquity factor [6]:

(10)

Where eR is the earth radius and Ih is the ionospheric

shell height (taken to be 350 km). Multiplying the right

hand side of equation (9) by the obliquity factor given in

equation (10) produces an expression for the ionospheric

decorrelation error:

(11)

It is important to note that equation (11) was derived for

the LAAS program where the terminal navigation

problem involves a straight approach (similar to the case-

III approach shown in figure 1). For that landing

configuration, ionospheric gradients can only exist in the

x-direction. However in the case-I approach, the UAV is

separated from the carrier in both the x and y directions

yIxIur vigyOBvigxOBI

2,

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1001000

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(see figure 2), allowing for ionospheric gradients to exist

in two dimensions. To account for this fact, equation (11)

is expanded to include two vig states per satellite:

xvig and yvig .

(12)

Similar to the tropospheric decorrelation error, the

residual ionospheric error for all satellites can be placed

in one matrix. Recognizing that ionospheric gradients can

exist in both the x and y directions, these matrices

become:

(13)

(14)

Where J is the familiar transformation matrix which

converts single difference measurements into double

difference measurements.

(15)

Similar expressions exist for the L2 ionospheric

decorrelation error.

(16)

(17)

Recall that double difference carrier phase measurements

are used for positioning. The model for the range term

given in equations (5) and (6) can be written in matrix

form for all satellites as:

(18)

Using equations (8), (13), (14), (15), (16), (17) and (18)

the double difference carrier phase measurement model is

given by:

(19)

atmosLLLLur ξmmNNxξ 2121

Where atmosξ is a subset of the state vector corresponding

to the atmospheric states n , xvig and yvig .

(20)

Note that the measurement vector given in equation (19)

consists of the geometry-free measurement GFz and

L1 and L2 double difference carrier phase measurements.

It is important to realize that GFz is only included in

the measurement model when the Kalman filter is

initialized. After the geometric redundancy snapshot,

only carrier phase measurements can be used for relative

positioning.

The state transition matrix is given by:

(21)

Where kGPS ,x is the GPS state vector at time k and

kGPS ,w is the associated process noise at time k.

00000000

00000000

0000000

000e1Iσ0000

0000e1Iσ000

00000000

00000000

0000000I

w

τT

2

Δυ

τT

2

Δυ

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)(cov GPS

I0000000

0I000000

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00000I00

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(22)

Where T is the sampling time and is the time constant

associated with the L1 and L2 multipath models. Notice

that the state transition matrix assumes all state variables

are constant from one time step to the next with the

exception of the multipath states. While this model is

exact for the integer states and sufficient for the

atmospheric states, it is not adequate for the position state.

It is obvious that the UAV will move from one

measurement epoch to the next, and this is not captured in

the dynamic model. In order to account for the lack of a

dynamic model for the UAV, we put infinite process

noise on the position state. The process noise covariance

matrix can therefore be written as:

(23)

Where is the standard deviation of the single

difference carrier phase multipath driver noise.

BASELINE GPS ALGORITHM SIMULATION

The performance of the GPS architecture can be evaluated

through a covariance analysis. Parameters used to carry

out the simulation are given below in table 1.

Table 1: Simulation parameters

Parameter Simulation Value

Mission Location Atlantic Ocean

Satellite Constellation 30 SV almanac

Single difference code sigma (σΔρ)

0.5 m

Single difference Carrier sigma (σΔφ)

1 cm

Airframe multipath Time constant (τa)

20 sec

Shipboard multipath Time constant (τs)

60 sec

Aircraft elevation mask 5 deg

Shipboard elevation mask 7.5 deg

Uncertainty in vertical ionospheric gradient (σvig)

4 mm/km

Uncertainty in refractivity Index (σΔn)

10

Integrity risk 10-7

The performance of the navigation architecture can be

examined for a single approach on the basis of

availability. A given approach is said to be available if

the accuracy and integrity requirements are satisfied at

each point along the approach. For the autonomous

shipboard landing application, accuracy and integrity

requirements are allowed to vary as a function of distance

to touchdown, the logic being that these requirements will

become more stringent as the UAV comes closer to

completing its mission. As we proceed, a hypothetical

example of required vertical positioning performance will

be defined as a function of distance to touchdown. This is

done because the vertical positioning error is generally

what results in an unavailable approaches.

Three factors must be considered when determining

where to fix the cycle ambiguities: pre-filtering duration,

the dynamic nature of the performance drivers and

ionospheric and tropospheric robustness. Of course, we

wish to pre-filter as long as possible since this will

improve cycle ambiguity resolution. However note that

before reaching the initial fix point, the UAV must rely on

a floating solution for positioning. Since the performance

criteria tighten as the UAV comes closer to landing, there

is a point where the floating solution will not satisfy the

requirements, rendering the approach unavailable. Lastly,

the Kalman filter propagation must be robust to the

ionospheric and tropospheric decorrelation error models.

If the baseline between the UAV and ship becomes too

large, the fidelity in the error models diminishes. In order

to accommodate these issues, the fixing point shown in

figure 4 was selected. This point is approximately 2

nautical miles away from the ship and represents a good

compromise for the issues described above.

Figure 4: Initial fix point

Using the requirements given in table 1 and the initial fix

point shown in figure 4, we can determine the vertical

positioning performance for a single approach. Figure 5

depicts the vertical positioning error ( v ) under various

assumptions regarding satellite blockage. The black

curve is the actual v that would be observed in the case-

I approach. Obviously, this approach would be

unavailable since v exceeds the accuracy requirement.

However, this case illustrates two interesting effects of

the satellite blockage.

Figure 5: Vertical positioning performance of baseline

GPS architecture

Firstly, signal obstruction has an adverse effect on the

pre-filtering process. As soon as a satellite is blocked,

any pre-filtering information that has been obtained is

lost. Consequently, the pre-filtering process must be re-

started once that satellite comes back online. The net

effect is a reduction in the observability of the L1 and L2

cycle ambiguities, which reduces the number of fixed

ambiguities. This is precisely what is observed in figure

5. The red curve plots v assuming that there is no

satellite blockage. It is apparent that v starts out with a

significantly lower value compared to the black curve,

indicating a higher number of fixed ambiguities.

Additionally, satellite loss in the final case-I bank leads to

a negative impact on vertical position error. The blue

curve indicates that the approach would be available if

there was no satellite blockage in the final bank. These

observations suggest two methods to improve availability.

One method to improve availability is to address the pre-

filtering limitation. While this is a viable option, it will

not be considered in this work. The other option is to

address the problem of satellite blockage through the

introduction of an inertial navigation system and a “dual-

track” fixing algorithm. Each of these approaches will

now be discussed individually.

DUAL TRACK FIXING ALGORITHM

The dual track fixing algorithm was developed to contend

with satellite blockage for aerial refueling [4]. In this

work, we modify the basic structure to take better

advantage of the Kalman filter propagation. The basic

premise of the dual track fixing algorithm is to

continually attempt to fix the current satellite set within

the integrity risk requirement. In this way, we ensure the

use of the best available satellite geometry for

positioning. This algorithm is shown schematically in

figure 6.

Figure 6: Dual track fixing algorithm

The dual track fixing algorithm begins in the same

manner as the baseline GPS navigation architecture

shown in figure 3. Pre-filtered widelanes are used with

the available carrier phase measurements to execute one

geometric redundancy snapshot, resulting in a floating

covariance matrix,

kfltP , . Integer fixing is conducted

next with high integrity which outputs a fixed covariance

EINB

Eu

BI

Eu

BN

UAV

n

E

Tv

NB

Eu

NEEIUAV

n

E

ωRFωF

gfR

E

v

0FRF

0ωω

E

v

2

2

0

0

0

ωRωF

gfR

g

b

b

E

v

F

g

b

b

E

v

EINBBI

Eu

BN

gm

a

g

UAV

n

E

gm

a

g

UAV

n

E

INS

matrix,

kfixP , . After the initialization step, there are two

options. First, a Kalman filter is used to propagate

kfltP , forward and re-execute the integer fixing step,

resulting in a new fixed covariance,

kfixnewP ,_ . This is

the red track shown in figure 6. The other option is to

simply propagate the previously fixed covariance matrix

kfixP , forward using the Kalman filter propagation

equations. This is the blue track shown in figure 6. If the

vertical positioning error is smaller using the newly fixed

covariance, then the primary positioning track will switch

to the auxiliary track. If there is no improvement in v ,

then no switch is made. This process is repeated at each

epoch and hence provides a means of recovering blocked

satellites. An important issue must be addressed here

related to integrity. The integer fixing step is conducted

in accordance with a probability of incorrect fix, which is

budgeted from the allotted integrity risk. Once the cycle

ambiguities have been resolved within the integrity risk,

any further fixing will cause an integrity breach. The

beauty of the dual track fixing algorithm is that it does not

take advantage of any information obtained from prior

fixing attempts. Hence, cycle ambiguities are reacquired

within the integrity risk requirement.

GPS/INS INTEGRATION

GPS and inertial navigation systems can be coupled using

a variety of integration schemes. These can range from

the simple loosely coupled integration, to the complex

ultra-tightly coupled mode in which the INS directly aids

the GPS tracking loops [6]. In this work, we adapt a

variant of the tightly coupled integration to the shipboard

landing scenario. This integration scheme is attractive

because it provides a method to combine GPS and INS

states (including cycle ambiguity states) into one

centralized Kalman filter [7]. To see how this is done,

consider first the fundamental equations of inertial

navigation.

For an inertial navigation system, one can derive

continuous time dynamic models for the velocity and

attitude of the roving vehicle [8]. In state space form,

these equations are given by:

(24)

Where UAV

n

Ev is the ground velocity of the UAV, E is

the attitude of the UAV (roll, pitch, and yaw), EI

ω is the

rotation rate of the earth, NEω is the rotation rate of the

UAV navigation frame relative to earth, NB

R is the

transformation matrix from the UAV navigation frame to

body frame, f is the specific force measured by the

accelerometers, g is the gravity vector, BI

ω is the inertial

angular velocity of the UAV measured by the gyroscopes,

EuF is the transformation matrix which translates the

instantaneous body-to-navigation rotation rate, BNω ,

into Euler angle rotation rate [9], and Tv2F is the matrix

which translates the UAV velocity, v, into the navigation

frame rotation rate, NEω [8].

In addition to these fundamental INS states, one must also

include states which model the gravity vector as well as

inertial sensor errors such as scale factor errors and

misalignment errors. Adding these states to the dynamic

model given in equation (24) results in:

(25)

where the dynamic matrix is defined as [7]:

gm

a

g

Euv2T

NB

Eu

gm2v

BNNEEI

I0000

0I000

00I00

00F0FRF

FR00ωω

F

τ

τ

τ

INS

2

INSLinINS

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

W

g

b

b

E

v

F

g

b

b

E

v

gm

a

g

UAV

n

E

gm

a

g

UAV

n

E

,

UAV

s

ship

ss xxx

The states gb and ab model the gyroscope and

accelerometer random bias and are described by first

order Gauss Markov processes with time constants g

and a , respectively. In addition, the gravity vector will

change in magnitude and direction due to the

inhomogeneous mass distribution of the earth. These

effects are captured in the state gmg , also modeled as

first order Gauss Markov with a time constant gm .

Notice that equation (25) is a continuous time non-linear

dynamic model for an inertial navigation system. For

simulation purposes, it is customary to simulate such a

system by using a linearized Kalman filter. This involves

linearizing equation (25) about some nominal trajectory

using either a Taylor series expansion or a perturbation

method. The details of this linearization process will not

be carried out in detail here, but can be found in

[7],[8],[9]. Using a linearized Kalman filter recasts the

state estimation problem in terms of estimating deviations

of the actual state from the reference trajectory. For

example, instead of estimating the UAV ground velocity, UAV

n

Ev , one would be estimating the deviation of the

UAV velocity from the reference value, UAV

n

Ev .

Hence, the linearized INS equations become:

(26)

where the linearized dynamic matrix is defined as:

gm

a

g

BNNI

v2T

gm2v

BN

I0000

0I000

00I00

00RωF

FR0f0

F

τ

τ

τ

LinINS ,

The process noise term, INSW , is inserted here to factor

in random disturbances such as scale factor errors,

misalignment errors, and random sensor noise in the

accelerometer and gyroscopes. This term can be written

as [7]

gm

a

g

g

BIg

is

BIg

f

BN

a

a

is

a

f

BN

η

η

η

νωMωSR

νfMfSR

W

δδ

δδ

INS

where a

fS and g

fS are the scale factor error matrices

for the accelerometer and gyroscope, respectively, a

isM

and g

isM are the misalignment matrices associated with

the accelerometer and gyroscope, aν is the random

sensor noise in the accelerometer, gν is the random

sensor noise in the gyroscope, and gη , aη , and gmη are

the driving white noise processes for the Gauss-Markov

models.

INTEGRATION MECHANISM

In order to tie the GPS and INS dynamic models together,

a relationship between the position state estimated with

GPS and the velocity state estimated with the INS must be

derived. This type of integration was first implemented in

[7] for a stationary reference station. The shipboard

landing scenario is different because the reference station

(aircraft carrier) is mobile.

Let the position of the UAV relative to the ship be

expressed in the ship local level frame, S.

(27)

dt

d

dt

d

dt

d UAV

s

Sship

s

S

s

Sxxx

UAV

s

S

s

E

UAV

s

E

ship

s

S

s

E

ship

s

E

s

S

dt

d

dt

d

dt

d

xωx

xωxx

s

S

s

EUAV

s

Eship

s

Es

S

dt

dxωvv

x

s

S

s

EUAV

a

EASship

s

Es

S

dt

dxωvRv

x

GPS

ship

ss

ship

s

INS

GPS

s

INS

δ

W

ωxv

W

η

x

η

F00

0ωC

00F

η

x

η

GPS

s

INS

LinGPS,

ship

s

E

LinINS,

0000000

0000000

000000

000I/000

0000I/00

0000000

0000000

F

0

, τ

τ

LinGPS

0000RC AS

Where sx is the relative position vector between the

UAV and ship expressed in the S frame, ship

sx is the

absolute position of the ship, and UAV

sx is the absolute

position of the UAV expressed in the S frame.

Differentiating both sides of equation (27) in the S frame

results in:

(28)

Now expand the right hand side of equation (28) in terms

of derivatives in the earth frame, E.

(29)

Where S

s

Eω is the angular velocity of the ship body

frame relative to the earth frame.

Using the definition of velocity, we arrive at:

(30)

However, the velocity that we estimate from the IMU is

the ground velocity of the UAV expressed in the UAV‟s

navigation frame. Therefore, we must use a

transformation matrix to transform UAV

n

Ev into

UAV

s

Ev .

(31)

Where AS

R is a rotation matrix from the UAV

navigation frame to the ship navigation frame.

Before equation (31) can be used in the dynamic model, it

must be put in a differential form using the same process

as in deriving equation (26). The resulting link between

GPS and INS states can be written as:

(32)

In order to incorporate the GPS dynamic model given in

equation (22) with the INS dynamic model given in

equation (26), it must be put in continuous form.

Furthermore, we must only consider the subset of the GPS

state transition matrix which corresponds to all states

excluding the position state. The reason for this is that

equation (32) already provides the dynamic model for the

relative position vector. Therefore, the appropriate

dynamic model for the GPS states is given by:

(33)

Equations (26), (32) and (33) can be put together in one

state space model, given below in equation (34).

(34)

C is a matrix defined as:

(35)

Here INSη is the INS state vector given in equation (26)

and GPSη is the subset of the GPS state vector given in

equation (12) which consists of all GPS states excluding

the relative position state. It is also worth mentioning that

since we are using a linearized Kalman filter, the GPS

states must also be written in terms of deviations from the

nominal trajectory. However, since the GPS dynamic

model is already linear, the structure of the state transition

matrix remains unchanged.

Also notice that the process noise term on the relative

position state involves knowing distributions for the

errors in the ship velocity and ship angular velocity. In

this work, we chose to neglect these terms and hence

assume there is no process noise on the relative position

state.

ship

s

E

s

s

ship

s

Eu

a

ASship

ss

ωx

xωvRvx

L2Δυ,

L1Δυ,

Δz

GPS

INS

GPS

ν

ν

ν

ξ

ξH0

υ

υ

zGF

2

1

L

L

GF

To summarize, equations (34) and (35) provide an

equation for the linearized dynamic model of the

GPS/INS integrated architecture. As desired, all GPS and

INS states have been put in one state vector which can be

estimated using a linearized Kalman filter. Before

showing results, note that the measurement model is the

same as that given in equation (12) since the only external

measurements are the GPS measurements. Hence, we

obtain:

(35)

Where GPSH is the GPS observation matrix given in

equation (12), INSξ is the INS state vector given in

equation (26) and GPSξ is the GPS state vector given in

equation (12).

GPS/INS SIMUATION RESULTS

The GPS/INS algorithm described above is applied to the

shipboard landing scenario. Vertical positioning

performance is compared under various combinations of

GPS, INS and dual track. The results are given below in

figure (7) for one approach.

Figure 7: Vertical positioning performance for GPS/INS

navigation architecture with satellite visibility history

Notice from figure 7 that the inclusion of INS does not

significantly improve the positioning performance for this

geometry. However, including the dual track fixing

algorithm improves v substantially and it is evident that

there is switching taking place. Nevertheless, the same

conclusion can be reached with the dual track fixing

algorithm: inclusion of the inertial navigation system does

not appreciably improve positioning performance.

There are two reasons why the INS does not show

improvement for the approach depicted in figure 7. First,

the simulation uses a large almanac of 30 SVs. Secondly,

the satellite visibility plot given in figure 7 shows that

there is not a high degree of satellite blockage. At most,

one satellite is lost which drops the number of SVs to 7.

With such a high number of satellites, it is reasonable to

suspect that the INS will not drastically improve

positioning performance for this particular approach.

However, all approaches have not been investigated and it

is quite possible that the INS will improve positioning

performance in these situations. Further study is required

and will be conducted in future work.

In order to really observe the benefit of this tightly

coupled architecture, consider an application which

involves severe sky blockage. For example, this could be

an aerial refueling mission or an urban canyon navigation

problem. To simulate this, we use the standard do229

almanac [12] and introduce a severe blockage situation.

Figure 8: Positioning performance in presence of severe

sky blockage with satellite visibility history

Notice from the satellite visibility history plot in figure 8

that the number of visible satellites drops to 3. Therefore,

the baseline GPS navigation algorithm will not be able to

provide a positioning solution and the approach will be

unavailable. Once the INS is included, the navigation

system is able to coast through this period of satellite

blockage as evidenced by the black curve in figure 8.

However, v is still above the accuracy requirement and

the approach remains unavailable. The inclusion of the

dual track fixing algorithm shows that a switch was made

just before dropping to 3 SVs (green curve). From here

the INS coasts through the outage and proceeds to the

touchdown point. We can see that in this situation it is

the combination of the INS and dual track fixing

algorithm which is able to render a previously unavailable

approach to an available state.

CONCLUSION

In this work an integrated GPS/INS navigation

architecture was developed for the shipboard landing

application. Using a novel tightly coupled approach with

the dual track fixing algorithm, it was shown for a single

landing approach that the INS does not result in a

significant improvement in positioning performance.

Again it must be restated that this result only applies to a

single approach. Future investigation of all approaches

will have to be conducted to determine the impact of an

inertial navigation system. However, it was shown that

this architecture is quite useful in situations with severe

satellite blockage. It was shown that the coasting

capability of the INS and satellite recovery capability of

the dual track algorithm were able to bring a previously

unavailable approach to an available state.

ACKNOWLEDGMENTS

We wish to thank our research sponsors who‟s generous

support has allowed us to continue to pursue this research

effort.

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